let G1 be _Graph; :: thesis: for G2 being DLGraphComplement of G1
for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 & v1 is isolated holds
( v2 .inNeighbors() = the_Vertices_of G2 & v2 .outNeighbors() = the_Vertices_of G2 & v2 .allNeighbors() = the_Vertices_of G2 )
let G2 be DLGraphComplement of G1; :: thesis: for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 & v1 is isolated holds
( v2 .inNeighbors() = the_Vertices_of G2 & v2 .outNeighbors() = the_Vertices_of G2 & v2 .allNeighbors() = the_Vertices_of G2 )
let v1 be Vertex of G1; :: thesis: for v2 being Vertex of G2 st v1 = v2 & v1 is isolated holds
( v2 .inNeighbors() = the_Vertices_of G2 & v2 .outNeighbors() = the_Vertices_of G2 & v2 .allNeighbors() = the_Vertices_of G2 )
let v2 be Vertex of G2; :: thesis: ( v1 = v2 & v1 is isolated implies ( v2 .inNeighbors() = the_Vertices_of G2 & v2 .outNeighbors() = the_Vertices_of G2 & v2 .allNeighbors() = the_Vertices_of G2 ) )
assume A1: ( v1 = v2 & v1 is isolated ) ; :: thesis: ( v2 .inNeighbors() = the_Vertices_of G2 & v2 .outNeighbors() = the_Vertices_of G2 & v2 .allNeighbors() = the_Vertices_of G2 )
then v1 .allNeighbors() = {} by GLIB_000:113;
then A2: ( v1 .inNeighbors() = {} & v1 .outNeighbors() = {} ) ;
hence A3: v2 .inNeighbors() = (the_Vertices_of G2) \ {} by A1, Th60
.= the_Vertices_of G2 ;
:: thesis: ( v2 .outNeighbors() = the_Vertices_of G2 & v2 .allNeighbors() = the_Vertices_of G2 )
thus v2 .outNeighbors() = (the_Vertices_of G2) \ {} by A1, A2, Th60
.= the_Vertices_of G2 ; :: thesis: v2 .allNeighbors() = the_Vertices_of G2
hence v2 .allNeighbors() = the_Vertices_of G2 by A3; :: thesis: verum